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Stable Matching Games: Manipulation via Subgraph Isomorphism

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Abstract

In this paper we consider a problem that arises from a strategic issue in the stable matching model (with complete preference lists) from the viewpoint of exact-exponential time algorithms. Specifically, we study the Stable Extension of Partial Matching (SEOPM) problem, where the input consists of the complete preference lists of men, and a partial matching. The objective is to find (if one exists) a set of preference lists of women, such that the men-optimal Gale–Shapley algorithm outputs a perfect matching that contains the given partial matching. Kobayashi and Matsui (Algorithmica 58:151–169, 2010) proved this problem is NP-complete. In this article, we give an exact-exponential algorithm for SEOPM running in time \(2^{{\mathcal {O}} (n)}\), where n denotes the number of men/women. We complement our algorithmic finding by showing that unless Exponential Time Hypothesis (ETH) fails, our algorithm is asymptotically optimal. That is, unless ETH fails, there is no algorithm for SEOPM running in time \(2^{o(n)}\). Our algorithm is a non-trivial combination of a parameterized algorithm for Subgraph Isomorphism, a relationship between stable matching and finding an out-branching in an appropriate graph and enumerating all possible non-isomorphic out-branchings. Our results cover both the cases when the preference lists are strict and complete, and when they are strict but possibly incomplete.

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Notes

  1. 1.

    In the stable roommates problem, the matching market consists of agents of the same type, as opposed to the market modeled the stable marriage problem that consists of agents of two types, men and women. Roommates assignments in college housing facilities is a real world application of the stable roommates problem.

References

  1. 1.

    Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pp. 295–304 (2007)

  2. 2.

    Roth, A.E., Oliveira Sotomayor, M.A.: Two-Sided Matching: A Study in Game Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1990)

  3. 3.

    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)

  4. 4.

    Gusfield, D., Irving, R.W.: The Stable Marriage Problem-Structure and Algorithm. MIT Press, Cambridge (1989)

  5. 5.

    Manlove, D.F.: Algorithmics of matching under preferences, volume 2 of Theoretical Computer Science. World Scientific (2013)

  6. 6.

    Kobayashi, H., Matsui, T.: Successful manipulation in stable marriage model with complete preference. IEICE Trans. Inf. Syst. E92–D(2), 116–119 (2009)

  7. 7.

    Kobayashi, H., Matsui, T.: Cheating strategies for the Gale–Shapley algorithm with complete preference lists. Algorithmica 58, 151–169 (2010)

  8. 8.

    Bredereck, R., Chen, J., Faliszewski, P., Guo, J., Niedermeier, R., Woeginger, G.J.: Parameterized algorithmics for computational social choice: nine research challenges. Tsinghua Sci. Technol. 19(4), 358–373 (2014)

  9. 9.

    Marx, D., Schlotter, I.: Parameterized complexity and local search approaches for the stable marriage problem with ties. Algorithmica 58(1), 170–187 (2010)

  10. 10.

    Marx, D., Schlotter, I.: Stable assignment with couples: parameterized complexity and local search. Discret. Optim. 8(1), 25–40 (2011)

  11. 11.

    Cseh, A., Manlove, D.F.: Stable marriage and roommates problems with restricted edges: complexity and approximability. In: Proceedings of the 8th International Symposium on Algorithmic Game Theory (SAGT), volume 9347 of LNCS, pp. 15–26 (2015)

  12. 12.

    Beyer, T., Hedetniemi, S.M.: Constant time generation of rooted trees. SIAM J. Comput. 9(4), 706–712 (1980)

  13. 13.

    Otter, R.: The number of trees. Ann. Math. 49(3), 583–599 (1948)

  14. 14.

    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative sets of product families. In: Proceedings of 22nd Annual European Symposium on Algorithms (ESA), volume 8737 of LNCS, pp. 443–454. Springer (2014)

  15. 15.

    Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S., Raghavendra Rao, B.V.: Faster algorithms for finding and counting subgraphs. J. Comput. Syst. Sci. 78(3), 698–706 (2012)

  16. 16.

    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Berlin (2010)

  17. 17.

    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

  18. 18.

    Amini, O., Fomin, F.V., Saurabh, S.: Counting subgraphs via homomorphisms. J. Discret. Math. 26(2), 695–717 (2012)

  19. 19.

    Impagliazzo, R., Paturi, R.: The complexity of k-SAT. In: Proceedings of 14th IEEE Conference on Computational Complexity, pp. 237–240 (1999)

  20. 20.

    Gale, D., Oliveira Sotomayor, M.A.: Ms. Machiavelli and the Gale–Shapley algorithm. Am. Math. Mon. 92(4), 261–268 (1985)

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Correspondence to Sanjukta Roy.

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Celebrating the 60th Birthday of Gregory Gutin.

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Gupta, S., Roy, S. Stable Matching Games: Manipulation via Subgraph Isomorphism. Algorithmica 80, 2551–2573 (2018). https://doi.org/10.1007/s00453-017-0382-5

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Keywords

  • Stable matching
  • Gale–Shapley algorithm
  • Suitor graph
  • Subgraph isomorphism
  • Exact-exponential time algorithms
  • Manipulation